Overview

The definite integral calculates the accumulated area under a curve over an interval. This topic introduces the concept using Riemann sums, then applies the Fundamental Theorem of Calculus to evaluate integrals exactly using antiderivatives.

Key Concepts and Structures

• Definite Integral: \int_a^b f(x) dx gives the net area between f(x) and the x-axis from x = a to x = b
• Riemann Sums: Approximate the integral using rectangles:
• Left endpoint, right endpoint, midpoint rules
• Fundamental Theorem of Calculus (Part 1):
  • If F is an antiderivative of f, then \int_a^b f(x) dx = F(b) - F(a)
• Fundamental Theorem (Part 2):
  • \frac{d}{dx} \int_a^x f(t) dt = f(x)
• Properties of Definite Integrals:
  • \int_a^a f(x) dx = 0
  • \int_a^b f(x) dx = -\int_b^a f(x) dx
  • \int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx
• Units: The result of a definite integral has units of (y-units) × (x-units), such as ft·sec

Quick Tip

Practice Problems

1. Evaluate \int_1^4 (2x + 1) dx
   Show Solution

   Antiderivative of 2x + 1 is x^2 + x

   F(4) = 16 + 4 = 20; F(1) = 1 + 1 = 2

   Final Answer: 20 - 2 = 18

2. Let F(x) = \int_0^x \cos(t) dt. Find F'(\pi/3)
   Show Solution

   By the Fundamental Theorem Part 2: F'(x) = \cos(x)

   Final Answer: \cos(\pi/3) = 1/2